A question inspired by <https://mathoverflow.net/questions/25922/is-the-euler-characteristic-a-birational-invariant>:

As remarked in Mike Roth's answer to the above linked question, if $X$ and $Y$ are smooth projective varieties in characteristic zero that are birational, then there is a smooth $Z$
with morphisms $p: Z \rightarrow X$ and $q: Z \rightarrow Y$ that are both birational isomorphisms. 

Question: Is it possible to arrange for at least one of $p$ and $q$ that the locus in $Z$ where the morphism fails to be an isomorphism is of codimension at least $2$? 

Here it might be useful to assume that ${\rm dim}X={\dim Y} \geq 3$.